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Abstract

Magnetic Resonance Elastography (MRE) is an emerging method for non-invasive breast cancer
screening. It takes the MRI displacement data output and reconstructs the internal stiffness
distribution, where cancerous tissue is approximately five to ten times stiffer than healthy breast
tissue. Hence, MRE offers a high contrast solution to this diagnostic problem.
Current MRE methods for reconstructing stiffness use forward simulation based optimization
methods that are highly non-linear, non-convex and very heavy computationally. This research
develops integral-based inverse problem solutions that reformulate the underlying differential
equations in terms of integrals of MRI measured displacement data, and this transforms the
problem into a linear, convex optimization. All derivative terms in the formulation are removed
by special choice of integration limits, so no smoothing or filtering of the input data is required.
The resulting equations can easily be solved by linear least squares requiring very minimal
computation.
1D inverse algorithms were developed to provide a proof of concept of the integral-based
method. Initially, the complete compressible 2D Navier's equations were used to develop the 2D
inverse methods. Reasonable results were achieved with the algorithm successfully identifying a
1cm by 1cm tumour with up to 10% noise, data resolution of 20 measured points per cm and
actuation frequencies of 100Hz.
However, for the same input data set, a simplified incompressible 2D model was used as the
basis for the final proposed inverse algorithm. This approach significantly improved results by
removing ill-conditioned terms from the original formulation. For a 1cm by 1 cm tumour,
accurate results were obtained with up to 40% noise, a range of actuation frequencies and very
low data resolution of the order of 2 measured points per cm. These results thus indicate that
more crude and less expensive data measurement systems could be used to obtain good results.
The methods developed can be readily extended to 3D by applying a similar incompressible
integral formulation to the 3D Navier’s equations.